Difficulty: Intermediate | Prerequisites: conic sections (ellipses, hyperbolas, parabolas), basic 3D coordinate systems
Quadric surfaces are the 3D analogues of conic sections. Just as ellipses, parabolas, and hyperbolas are defined by second-degree equations in two variables, quadric surfaces are defined by second-degree equations in three variables. Being able to identify and sketch these surfaces is a prerequisite for working with double and triple integrals later, where the region of integration is often bounded by one or more quadric surfaces. If you are comfortable with conic sections in 2D, the main new skill here is reading an equation and picturing which 3D shape it describes.
There are six standard quadric surfaces (plus cylinders) that you need to recognise from their equations. The key technique is taking traces: slice the surface with planes parallel to the coordinate planes, see what 2D curve you get, and use those slices to identify and sketch the shape.
Quadric surface
A surface in R3 defined by a second-degree polynomial equation in x, y, and z.
Think of it as the 3D version of a conic section.
Trace
The cross-section you get when you slice a surface with a plane (usually x = k, y = k, or z = k for some constant k).
In simple terms, set one variable equal to a number and see what 2D curve remains.
Ellipsoid
A closed surface where all traces are ellipses. Equation: x2/a2 + y2/b2 + z2/c2 = 1. When a = b = c, it is a sphere.
Cone
A surface where one variable equals the square root of a sum of squares of the other two. Equation: z2/c2 = x2/a2 + y2/b2. Horizontal traces are ellipses; vertical traces in x = k and y = k are hyperbolas (or pairs of lines when k = 0).
Elliptic paraboloid
A bowl-shaped surface. Equation: z/c = x2/a2 + y2/b2. Horizontal traces are ellipses, vertical traces are parabolas. The variable raised to the first power indicates the axis of the paraboloid.
Hyperbolic paraboloid (saddle surface)
A saddle-shaped surface. Equation: z/c = x2/a2 - y2/b2. Horizontal traces are hyperbolas, vertical traces are parabolas opening in opposite directions.
Hyperboloid of one sheet
A connected surface that looks like a cooling tower. Equation: x2/a2 + y2/b2 - z2/c2 = 1. Horizontal traces are ellipses, vertical traces are hyperbolas. The axis of symmetry corresponds to the variable whose coefficient is negative.
Hyperboloid of two sheets
Two separate bowl-like pieces. Equation: -x2/a2 - y2/b2 + z2/c2 = 1. Two minus signs in the equation signal two sheets. Horizontal traces z = k are ellipses only when |k| > c.
Cylinder (in the quadric sense)
A surface whose equation is missing one variable entirely. The missing variable can take any value, so the 2D cross-section is extruded along that axis.
Write the equation in standard form (all terms on one side, constant on the other)
Count positive and negative squared terms
Check whether one variable appears only to the first power (paraboloid) or not at all (cylinder)
Take traces to confirm the shape
Ellipsoid: x2/a2 + y2/b2 + z2/c2 = 1
All traces are ellipses
Special case: a = b = c gives a sphere
Bounded in all directions
Cone: z2/c2 = x2/a2 + y2/b2
Horizontal traces are ellipses
Vertical traces in x = k or y = k are hyperbolas when k is not 0, and pairs of intersecting lines when k = 0
Vertex at the origin
Elliptic paraboloid: z/c = x2/a2 + y2/b2
Horizontal traces are ellipses
Vertical traces are parabolas
The variable to the first power tells you which axis the bowl opens along
Hyperbolic paraboloid: z/c = x2/a2 - y2/b2
Horizontal traces are hyperbolas
Vertical traces are parabolas
Saddle point at the origin; the surface curves up in one direction and down in the other
Sometimes called a "saddle" or "Pringles chip" shape
Hyperboloid of one sheet: x2/a2 + y2/b2 - z2/c2 = 1
Horizontal traces are ellipses
Vertical traces are hyperbolas
One connected piece; looks like a cooling tower or an hourglass with the waist opened out
The axis of symmetry is the variable with the negative coefficient
Hyperboloid of two sheets: -x2/a2 - y2/b2 + z2/c2 = 1
Two minus signs in the equation indicate two separate sheets
Horizontal traces z = k are ellipses only for |k| > c
Vertical traces are hyperbolas
Two separate bowl-like pieces opening in opposite directions along the positive-coefficient axis
\text{Ellipsoid: } \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\text{Cone: } \frac{z^2}{c^2} = \frac{x^2}{a^2} + \frac{y^2}{b^2}\text{Elliptic paraboloid: } \frac{z}{c} = \frac{x^2}{a^2} + \frac{y^2}{b^2}\text{Hyperbolic paraboloid: } \frac{z}{c} = \frac{x^2}{a^2} - \frac{y^2}{b^2}\text{Hyperboloid of one sheet: } \frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\text{Hyperboloid of two sheets: } -\frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1Ellipsoids model the shape of planets (Earth is an oblate ellipsoid, slightly squashed at the poles). Hyperboloids of one sheet are used in the design of cooling towers at power stations because the shape is strong under compression. Parabolic reflectors (satellite dishes, car headlights) are surfaces of revolution formed by rotating a parabola, closely related to elliptic paraboloids.
Students often confuse the hyperboloid of one sheet with the hyperboloid of two sheets. Count the minus signs: one minus sign in front of a squared term means one sheet, two minus signs means two sheets.
A cone is not the same as a paraboloid. The cone has the isolated variable squared (z2 = ...), while the paraboloid has it to the first power (z = ...).
Students sometimes think a cylinder must be circular. In multivariable calculus, a cylinder is any surface whose equation is missing one variable entirely, so it can be elliptic, parabolic, or hyperbolic.
The phrase "traces are ellipses" does not mean the surface is an ellipsoid. Cones and hyperboloids of one sheet also have elliptical horizontal traces.
⚠️ Expect questions that give you an equation and ask you to name the surface and describe its traces.
⚠️ Know how to sketch each surface from its equation. Traces are your main tool.
⚠️ Pay attention to which variable carries the negative sign or is to the first power. This determines the axis of symmetry and the orientation of the surface.
⚠️ Cylinders are easy marks if you remember to check whether a variable is missing from the equation.
True or false: An ellipsoid has hyperbolic traces.
False. All traces of an ellipsoid are ellipses.
Fill in the blank: A surface whose equation is missing one variable entirely is called a ___.
Cylinder.
True or false: The equation x2/4 + y2/9 - z2/16 = 1 describes a hyperboloid of two sheets.
False. One minus sign means one sheet. It is a hyperboloid of one sheet.
Fill in the blank: In the equation z/c = x2/a2 + y2/b2, horizontal traces are ___ and vertical traces are ___.
Ellipses; parabolas.
True or false: A cone and an elliptic paraboloid both have elliptical horizontal traces.
True.
Q: Identify the surface given by x2/9 + y2/4 + z2 = 1 and describe its traces.
A: This is an ellipsoid with a = 3, b = 2, c = 1. All traces (x = k, y = k, z = k) are ellipses.
Q: What surface does z = x2 + y2 describe? What are its traces?
A: An elliptic paraboloid (with a = b = 1, c = 1). Horizontal traces (z = k, k > 0) are circles; vertical traces are upward-opening parabolas.
Q: How do you tell a hyperboloid of one sheet from a hyperboloid of two sheets?
A: Count minus signs among the squared terms. One minus sign: one sheet. Two minus signs: two sheets.
Q: What type of surface is x2 - y2 = 1? (Note the missing variable.)
A: This is a hyperbolic cylinder extending along the z-axis, because z is absent from the equation entirely.
Quadric surfaces define the boundaries of integration regions in double and triple integrals (Chapters 15-16). Recognising the surface helps you choose the right coordinate system (cylindrical or spherical). Traces are also useful when studying level surfaces of multivariable functions, which reappear with partial derivatives and gradients.
quadric surface, ellipsoid, cone, elliptic paraboloid, hyperbolic paraboloid, saddle surface, hyperboloid of one sheet, hyperboloid of two sheets, cylinder, traces, cross-section, conic section, standard form, MATH 126, Calculus III, multivariable calculus, 3D surfaces, level surfaces